Defining Polar Coordinates and Differentiating in Polar Form
Interpret directed polar radii and use coordinate rates for tangent and second-derivative reasoning.
College Board Topic 9.7 · CED p. 172 (PDF p. 177) · CED effective Fall 2020; project course outline 2026–27.
A polar point uses a directed radius r and angle θ: , . A negative radius points in the opposite direction; in polar coordinates is the same Cartesian point as . Multiple angles and signed radii can name the same point. Use radians for calculus.
For , treat θ as a parameter. The product rule gives and . Consequently when the denominator is nonzero. The radial rate dr/dθ tells whether the directed radius is changing; it is not a Cartesian tangent slope.
For the second derivative, reuse the parametric formula: let , then . Equivalently it is . Here and . Check x′≠0. Horizontal/vertical tangent tests use both coordinate rates; when both vanish, examine the curve locally instead of dividing by zero.
Sketch by making an angle/radius table, converting selected points, and following the stated angle interval. A polar graphing window can confirm a proposed trace, but be alert to loops, negative r and repeated tracing.
Let's work through it
- For at , obtain , , . Why: Differentiate the radial function with respect to its angle.
- Compute , , so . Why: The product-rule coordinate derivatives determine Cartesian slope.
- Compute , , giving . Why: Use (y″x′−y′x″)/(x′)³ because x′ is nonzero.
Try it from memory
Pause & retrieve
Try it before you peek.
Write your answer and the reason from memory. Then compare with the worked answer. Your self-assessment is a study decision, not an automatically graded score.
Question 1
See every recall question with its answer
These references preserve each prompt and its answer for direct evidence links. Try the interactive questions above before opening the worked reasoning.
Retrieval 1
Worked answer and reason
The coordinate rates also contain sine and cosine factors.
Retrieval 2
Worked answer and reason
x=r cos θ=−2 and y=r sin θ=0.
Catch the tempting mistake
- Set dr/dθ=0 to find every horizontal tangent.
- Horizontal tangents depend on dy/dθ=0 with dx/dθ≠0, not only radial change.
Calculator workspace and limits
For calculus, set the angle selector to Radians. This scientific calculator evaluates arithmetic and functions. It does not graph arbitrary functions, find numerical roots, evaluate numerical derivatives, or compute definite integrals. Those tasks need an approved graphing tool; the May 2027 exam permits approved handheld graphing calculators and Bluebook’s built-in Desmos on calculator-required parts. A graph or decimal estimate helps you check a conclusion; your setup, hypotheses, units and justification still do the explaining.
Scientific calculator
This calculator requires JavaScript. Use the physics formula sheets and show your working manually.
+ − * / ^ ! · pi · e · sqrt · sin, cos, tan · asin, acos, atan · atan2(y,x) · ln, log · abs · exp · hypot(x,y). Use * between factors. Keep units in your written working.
Put it to work
- Unit 9 problem 19 · No calculator · analytic
- Unit 9 problem 20 · No calculator · analytic
- Unit 9 problem 21 · No calculator · analytic
- Unit 9 problem 22 · No calculator · graphical
For the teacher
- Have students compare a point represented with positive and negative radius. For second derivatives, assess the chain-rule denominator even when arithmetic is messy.
Glossary and deeper reading · 4 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Radian
A radian measures an angle by dividing the intercepted arc length by the circle’s radius. One full turn is 2π radians, so π radians equals 180 degrees. For example, an arc with length equal to its radius subtends an angle of one radian. The familiar derivative formulas for sine and cosine use radian inputs.
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Product rule
The product rule differentiates a product of two differentiable functions. It gives (fg)′=f′g+fg′, changing one factor at a time and adding the results. For example, the derivative of x sin x is sin x+x cos x. The product of the two derivatives is generally not the derivative of the product.
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Second derivative
A second derivative is the derivative of the first derivative. It measures how the first rate of change changes with the input; further repeated differentiation gives higher-order derivatives. For example, f(x)=x³ has f′(x)=3x² and f″(x)=6x. For position measured against time, the second derivative is acceleration.
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Vertical tangent
A vertical tangent is a tangent line parallel to the vertical axis, with no finite slope. A graph can have a vertical tangent even though its derivative does not exist as a finite real number at that input. For example, y=∛x has the vertical tangent x=0 at the origin as its nearby slopes tend to positive infinity. This case is separate from the finite-slope tangent approximation available at a differentiable point.